Regular readers of Nobel Intent know that one of my favorite topics to cover (and study) is computer simulation of real world phenomena, specifically statistical thermodynamic and molecular phenomena. Molecular simulations, and simulations in general, have always held one feature that no experimental work can match: in the simulation world. You get to make up the rules of the universe. Simply put, reality—on the molecular level—is too computationally expensive to simulate with anywhere near 100 percent accuracy. Since you get to choose the simplifications you use, you really can make a horse behave like a sphere if it will make the math easier.

Ever since Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller's seminal paper in 1953, where the equation of state of hard disks was computed, researchers have used simplified models of reality to understand the key physics that drive a system. After all, the thermodynamics that allow one to fully describe a system does not care how close to reality the model used is.

In a paper* set to be published in an upcoming edition of Langmuir, a trio of researchers from Washington University in St. Louis have looked at modeling the thermodynamics of Tetris pieces. Using the seven tetromino shapes—the shapes one encounters in the video game Tetris—as a proxy for molecules with a complex shape, the team modeled their adsorption on a flat surface to examine the self-assembly and thermodynamic properties of these systems. While no one will confuse the results of this work with an actual adsorption study of a mixture of complex molecules, this simplicity of the system allowed the trio to gain a much deeper insight into the nature of what is happening.

The simulations were carried out on a lattice, essentially a space that is discretized into little squares, and each piece—the square, the rod, the S-shape, the Z-shape, the L-shape, the J-shape, and the T-shape—occupied four squares each. Simulating the lattice surface in a grand canonical ensemble allowed them to model an open system; an initially blank surface that is exposed to an infinite reservoir of pieces at a fixed chemical potential, a property that is related to pressure in such a system.

The pieces were free to adsorb onto the surface, with the only constraints in the model being that no two pieces could overlap. As a result, there was no energy to minimize, so all phenomena was driven by the system's tendency to maximize its entropy. Using the results from these simulations, the authors were able to create some illuminating visuals.

For each pure fluid—all rods, or all S-pieces, etc.—the researchers calculated how the chemical potential (or pressure) of the external fluid affected the density of particles on the surface.

Even in these pure fluid systems, complex ordering phenomena emerged. The squares simply filled space, but aligned along the Cartesian coordinates. The rods, starting at medium densities, began aligning themselves to form small groups that, at higher densities, formed into large aligned squares. The S- and Z- shapes formed a herringbone pattern, whereas the L- and J-shapes would pair up, rotating to fit into one another with a 180o flip. All this was done simply by requiring that the shapes could not overlap—there is no energetic bonus to be gained from these higher order structures.

With such a simple model, it is possible to not only examine pure fluids, but mixtures of the various shapes as well. The researchers modeled two-, three-, four-, five-, six-, and even seven-component mixtures. From these, and the previous simulation results along with a handful of basic thermodynamics, a lot can be learned about the system.

For instance, squares and S-shapes form the most non-ideal mixture of the binary components pairs. If you create a mixture of about 75 mL of squares and 25 mL of S-shapes, the mixture will not have a volume of 100 mL, but would occupy about 105 mL. Through a mathematical computation of the second virial coefficient—a measure of how much two particles interact—the authors found that Z-shapes would preferentially mix with squares over rods, and that T-shapes most favorably interact with S-, Z-, and other T-shapes, and have the least favorable interaction with rods.

Taking the analysis further, the group computed the various solubilities of shapes in mixtures of other shapes. Again, using semi-classical thermodynamic analysis, they were able to measure the Henry's Law coefficients, which indicates how soluble one species is in a mixture of others. In this portion of the analysis, they demonstrated that squares are always the most soluble piece, while rods are the least. Squares turn out to be most soluble in the L- and J-shape fluids, and least soluble in S- and Z-shape fluids.

All of the work here was done in an attempt to examine the nature of complex molecules adsorbing at a surface. Even though it involved Tetris pieces getting places on a Battleship board, the amount of information derived is impressive, and shows how these simplistic models can give much deeper insights than would appear possible. The authors summarize their findings using the old chemistry adage of "like dissolves like." As they quip, "provided one has an expansive interpretation of 'like.'"

To me, this paper highlighted why I love simulation research: you can gain a much deeper understanding of phenomena when you are not burdened with the trappings of reality. Now, when I play a round of Tetris, I can think of how well the rod that is falling will mix with the other pieces I have strewn about the game board. As the authors conclude, "the results obtained to date may have some relevance to successful strategies for playing the Tetris computer game, but this has not been considered in detail."

Thanks to author Brian Barnes for bringing this paper and work to my attention.

11 Reader Comments

"The squares simply filled space, but aligned along the Cartesian coordinates"

why is this surprising? Or rather, how would the squares fill the space without aligning to the cartesian canvas? Since all the pieces seem to only rotate by 90deg, how does a square piece (or any other piece) not align with the cartesian coordinates?

Neat! They need to run this in 3D now, and create dipoles for the pieces, and we'll get even closer to chemistry. And I'm sure there are hundreds of little things that could be done after that to get even closer.

Can we take away the lesson that a particle's surface-to-volume ratio directly related to its solubility? Like whereas a square is best here, a sphere would be best in a real solution? I imagine adding dipoles might keep that from being universally true.

This is one of those studies you look at and think, "I wish I had thought of doing that." It looks like a lot of fun, and I can think of some applications. One might be able to say it's the new Ising model for extended structures, where steric interactions rule. (Certainly, the rod simulations look like Ising model results near the transition temperature.)

It would seem to me that such studies would be very useful for people studying liquid crystals (I don't follow that literature at all, but I wouldn't be surprised if people from that field would look at this study and think the results are obvious). It would probably make a great project for a simulation class, too.

From the standpoint of playing Tetris, IMHO, the results aren't that surprising, though. Yes, when playing Tetris, I have found that the rod pieces don't mix well and that I can almost always find a place to stick a square or a T-piece.

Hi everyone! I'm the Ars reader and submitter (Brian). I'm glad people find the work interesting. The paper should remain open-access, so anyone can download the full PDF. Thanks again to Matt for the nice write-up.

To brentK: yeah, the rod simulations are quite similar to basic liquid crystal studies of nematic/smectic phases. Pure rod phases are actually quite thoroughly studied in the past literature which we cited, but it's still neat to see how they interact with non-rod shapes.

We're currently performing simulations of larger polyominoes and investigating their phase transitions. No plans to move to 3D just yet (that's a whole different can of worms).

"...squares and S-shapes form the most non-ideal mixture of the binary components pairs. (...) the authors found that Z-shapes would preferentially mix with squares over rods, and that T-shapes most favorably interact with S-, Z-, and other T-shapes, and have the least favorable interaction with rods."

This is something that everyone who has played Tertis until the pieces fall before your closed eyes when you lay in bed knows.I forget who said "it is frequently found that a few hundred hours in the lab can often save a few hours in the library". This proves that a few houndred hours playing Tetris could save you from doing some pretty funny and interesting research. Whish I had done that instead of playing Tetris.....

Matt Ford / Matt is a contributing writer at Ars Technica, focusing on physics, astronomy, chemistry, mathematics, and engineering. When he's not writing, he works on realtime models of large-scale engineering systems.